A Fujita-type blowup result and low energy scattering for a nonlinear Schr\"o\-din\-ger equation

In this paper we consider the nonlinear Schr"o-din-ger equation $i u_t +\Delta u +\kappa |u|^\alpha u=0$\mathrm{.}Weprovethatif$\alpha <\frac {2} {N}$and$\Im \kappa <0$,theneverynontrivial$H^1$-solution blows up in finite or infinite time. In the case $\alpha >\frac{2}{N}$ and $\kappa \in {\mathbb C}$,weimprovetheexistinglowenergyscatteringresultsindimensions$N\ge 7$\mathrm{.}Moreprecisely,weprovethatif$ \frac {8} {N + \sqrt{ N^2 +16N }} < \alpha \le \frac {4} {N} $, then small data give rise to global, scattering solutions in $H^1$.